ROHF and ROKS (restricted open-shell SCF)¶

vibe-qc has two mean-field references for open-shell molecules: the unrestricted family (UHF / UKS) and the restricted open-shell family (ROHF / ROKS) documented here. Both describe the same radical or high-spin state; they differ in one decision: whether the α and β electrons are allowed independent spatial orbitals.

This page is the dedicated reference for ROHF/ROKS: when to pick them over UHF/UKS, how to run them, and what gradient/optimisation/post-SCF support exists. For the broader open-shell setup (multiplicity, charge) see molecules.md; for the Fock-build modes that apply to every SCF method see scf_modes.md.

When to use ROHF vs UHF¶

	UHF / UKS (unrestricted)	ROHF / ROKS (restricted open shell)
Spatial orbitals	independent α and β sets	one common set: closed + open + virtual
Spin purity	not a spin eigenfunction; ⟨S²⟩ deviates from `S(S+1)` (spin contamination)	spin-pure by construction: ⟨S²⟩ = `S(S+1)` exactly
Energy	variationally lowest single determinant	slightly higher than UHF (extra orbital constraint)
Best for	quick open-shell SCF, bond breaking / strong static correlation where symmetry breaking is wanted	spin-pure references for post-HF (ROHF-MP2 / ROHF-CC), a clean CAS starting point, and properties sensitive to spin contamination (hyperfine, spin densities)

Rule of thumb. Reach for ROHF/ROKS when spin contamination would corrupt the answer or the downstream method (a spin-pure CAS reference, an ROHF-CCSD(T) energy, a clean spin density). Reach for UHF/UKS when you want the lowest single-determinant energy, are breaking bonds, or need a large fast DFT relaxation. UHF’s ⟨S²⟩ is reported in every open-shell run so you can see the contamination directly.

Running ROHF / ROKS¶

The spin partition (n_alpha − n_beta = multiplicity − 1) is read from Molecule.multiplicity, exactly as for UHF. Use the dedicated drivers (run_rohf / run_roks) for a result object, or run_job for the full end-to-end run with output files + citations.

import vibeqc as vq

# Hydroxyl radical (doublet). See examples/molecular/input-oh-rohf.py.
oh = vq.Molecule(
    atoms=[vq.Atom(8, [0.0, 0.0, 0.0]),
           vq.Atom(1, [0.0, 0.0, 1.834])],   # bohr
    multiplicity=2,
)
basis = vq.BasisSet(oh, "6-31g*")

rohf = vq.run_rohf(oh, basis)        # spin-pure: <S^2> = 0.75 exactly
print(rohf.energy, rohf.s_squared)

# End-to-end (writes .out / .molden / .references / .bibtex):
vq.run_job(oh, basis="6-31g*", method="rohf", output="oh_rohf")

The Kohn-Sham counterpart is ROKS, the same Roothaan coupling with a spin-polarised XC potential:

roks = vq.run_roks(oh, basis, functional="pbe")          # spin-pure KS
vq.run_job(oh, basis="6-31g*", method="roks", functional="b3lyp",
           output="oh_roks")          # see examples/molecular/input-ch3-roks.py

ROKS supports LDA, GGA, meta-GGA, global-hybrid, and range-separated-hybrid functionals (B3LYP, PBE0, TPSS, r²SCAN, ωB97X, ωB97M-V, …). Double hybrids (B2PLYP, DSD-PBEP86, PWPB95) run through vibeqc.run_double_hybrid (spin-pure ROKS SCF + a semicanonical ROHF-MP2 doubles correction); a direct run_roks on a double-hybrid functional points you there because the SCF energy alone is incomplete. Convergence knobs live on ROHFOptions / ROKSOptions (max_iter, conv_tol_energy, conv_tol_grad, use_diis, level_shift, density_fit / aux_basis).

For degenerate ROKS terms, the KS driver averages the frontier shell instead of forcing one arbitrary component to be the SOMO. Linear OH(2Pi), for example, is represented with occupations 2, 2, 2, 1.5, 1.5, 0, ... across the pi pair; this preserves the restricted open-shell symmetry and prevents the larger-basis ROKS/PBE oscillation that an integer 2, 2, 2, 2, 1, 0, ... assignment produces.

Gradients, geometry optimisation, frequencies¶

ROHF has an analytic gradient (vibeqc.compute_rohf_gradient), so geometry optimisation and harmonic frequencies run at full speed:

opt = vq.run_job(oh, basis="6-31g*", method="rohf", optimize=True)
frq = vq.run_job(oh, basis="6-31g*", method="rohf", hessian=True)  # FD of the analytic gradient

ROKS geometry optimisation uses finite-difference forces for now (the analytic molecular XC-gradient term is pending), correct but slower; prefer UKS for large fast DFT relaxations. See geometry_optimization.md for the optimiser itself.

Beyond single points¶

ROHF/ROKS also drive:

Spin-pure CAS references: cas_reference="rohf" gives CASSCF/CASCI a spin-pure starting determinant (cleaner than UHF natural orbitals); see non_hf_solvers.md.
ROHF-reference CCSD / CCSD(T): run_job(method="ccsd(t)", ccsd_reference="rohf") runs the spin-orbital coupled-cluster kernel on the spin-pure reference; see ccsd.md.
Semicanonical ROHF-MP2: run_job(method="mp2", mp2_reference="rohf") (or vibeqc.run_rohf_mp2) computes second-order Møller-Plesset correlation on the spin-pure reference, including the Brillouin singles term that UMP2 lacks (Knowles et al. 1991). Needs a basis with a bundled RI auxiliary (cc-pVDZ / def2; Pople bases raise a clear error).
Atomization energies (run_job(atomization=True), with spin-pure free-atom references), PES scans (vibeqc.scan), and the molecular optimiser (vibeqc.molecular_optimize.optimize_molecule).
The ASE calculator: VibeQC(restricted_open=True) runs ROHF with analytic forces; see ase_integration.md.

Periodic ROHF¶

Restricted open-shell HF is available for periodic systems at the Γ-point and with full Brillouin-zone sampling (EWALD_3D), via the standalone drivers run_rohf_periodic_gamma_ewald3d and run_rohf_periodic_multi_k_ewald3d. A second standalone entry point, run_periodic_rohf_gpw, provides 3D Gamma-point HF with GPW Hartree J, per-spin exact K, and the Ewald one-electron/nuclear gauge. It uses the shared Roothaan loop to retain one spatial-orbital set and integer 2/1/0 occupations.

These drivers are not yet wired into the run_periodic_job dispatcher. The GPW route is not a GDF, BIPOLE, GAPW, or multi-k GPW implementation, and periodic ROKS (periodic XC) and electronic smearing are still pending. The smearing_alpha keyword on run_periodic_rohf_gpw controls nuclear-charge smoothing, not electronic Fermi smearing. See periodic_methods.md § 3.8 for the periodic SCF context.

Theory¶

ROHF uses Roothaan’s single effective Fock operator (Roothaan, Rev. Mod. Phys. 32, 179 (1960)). The MOs are partitioned into closed, open, and virtual blocks with integer occupations (2/1/0). A single effective Fock matrix is assembled from the per-spin Fα/Fβ via Coulson’s coupling so that one diagonalisation yields a spin eigenfunction with ⟨S²⟩ = S(S+1) exactly. ROKS reuses the identical coupling but with Fα/Fβ carrying the spin-polarised KS exchange-correlation potential and, for degenerate frontier terms, uniform fractional occupations within the open block. Its energy uses E_xc[ρα, ρβ] in place of full exact exchange. The roothaan_rohf_1960 citation fires automatically in any ROHF/ROKS run (including CAS jobs that use cas_reference="rohf").